Determining the Number of Factors in Two-way Factor Model of High-dimensional Matrix-variate Time Series: a White-noise Based Method for Serial Correlation Models

Qiang Xia, W.K. Li and Rubing Liang

doi:10.5705/ss.202025.0212

Abstract

This study proposes a novel two-way factor modeling framework for high-dimensional matrix

variate time series. Motivated by the objective of identifying white noise components, we develop two

ratio-based estimators leveraging the element-wise maximum norm and Frobenius norm of sample autocovariance matrices to determine the dimensions of row and column factor spaces. To reduce the impact

of cross-row and cross-column factor strength heterogeneity, the original matrix factor model is reparameterized as a reduced-form model containing only a row loading matrix or a column loading matrix. We

then investigate the refined ratio-based methods developed under this reparameterized framework. Under regularity conditions, we establish the theoretical properties of the proposed methods, demonstrating

their consistency in estimating the number of factors. Through Monte Carlo simulations and a real data

application, we validate the finite-sample performance of the proposed methods and compare them with

existing alternatives.

Key words and phrases: Matrix-variate time series, Max-type test, Sum-type test, Two-way factor models

Information

Preprint No.	SS-2025-0212
Manuscript ID	SS-2025-0212
Complete Authors	Qiang Xia, W.K. Li, Rubing Liang
Corresponding Authors	Qiang Xia
Emails	xiaqiang@scau.edu.cn

References

Bai, J. (2003). Inferential theory for factor models of large dimensions, Econometrica 71: 135– 171.
Bai, J. and Li, K. (2012). Statistical analysis of factor models of high dimension, The Annals of Statistics 40: 436–465.
Chang, J., Guo, B. and Yao, Q. (2018). Principal component analysis for second-order stationary vector time series, The Annals of Statistics 46: 2094–2124.
Chang, J., Yao, Q. and Zhou, W. (2017). Testing for high-dimensional white noise using maximum cross-correlations, Biometrika 104(1): 111–127.
Chen, E. Y. and Fan, J. (2023). Statistical inference for high-dimensional matrix-variate factor model, Journal of the American Statistical Association 118: 1038–1055.
Chen, E. Y., Tsay, R. S. and Chen, R. (2020). Constrained factor models for high-dimensional matrix-variate time series, Journal of the American Statistical Association 115: 775–793.
Fan, J., Guo, J. and Zheng, S. (2022). Estimating number of factors by adjusted eigenvalues thresholding, Journal of the American Statistical Association 117: 852–61.
Filipiak, K., Klein, D. and Vojtkova, E. (2018). The properties of partial trace and block trace operators of partitioned matrices, Electronic Journal of Linear Algebra 33: 3–15.
Forni, M., Hallin, M., Lippi, M. and Reichlin, L. (2005). The generalized dynamic-factor model: one-sided estimation and forecasting, Journal of the American Statistical Association 100: 830–840.
He, Y., Trapani, L., Kong, X. and Yu, L. (2023). One-way or two-way factor model for matrix sequences?, Journal of Econometrics 235: 1981–2004.
Lam, C. and Yao, Q. (2012). Factor modeling for high-dimensional time series: inference for the number of factors, The Annals of Statistics 40: 694–726.
Lam, C., Yao, Q. and Bathia, N. (2011). Estimation for latent factor models for highdimensional time series, Biometrika 98: 901–918.
Li, Z., Lam, C., Yao, J. and Yao, Q. (2019). On testing for high-dimensional white noise, The Annals of Statistics 47(6): 3382–3412.
Pan, J. and Yao, Q. (2008). Modelling multiple time series via common factors, Biometrika 95: 356–379.
Wang, D., Liu, X. and Chen, R. (2019). Factor models for matrix-valued high-dimensional time series, Journal of Econometrics 208: 231–248.
Yu, L., He, Y., Kong, X. and Zhang, X. (2022). Projected estimation for large-dimensional matrix factor models, Journal of Econometrics 229: 201–217.

Acknowledgments

The authors are grateful to the Co-Editor, the Associate Editor, and two anonymous referees

for their insightful comments and constructive suggestions, which significantly improved the

quality of this work. We also thank Prof. Lixing Zhu, Prof. Xianyang Zhang, and Dr. Zhaoxing

Gao for their valuable feedback on an earlier version of the manuscript. This research was

supported in part by the National Natural Science Foundation of China (No. 12171161) and

the National Statistical Scientific Research Projects (No. 2025LZ029).

Supplementary Materials

Numerical Studies

4.1

Simulation Experiments

We conduct simulation experiments to compare the proposed method with the eigenvalue ratio

Supplementary materials are available for download.

[1] Bai, J. (2003). Inferential theory for factor models of large dimensions, Econometrica 71: 135– 171.

[2] Bai, J. and Li, K. (2012). Statistical analysis of factor models of high dimension, The Annals of Statistics 40: 436–465.

[3] Chang, J., Guo, B. and Yao, Q. (2018). Principal component analysis for second-order stationary vector time series, The Annals of Statistics 46: 2094–2124.

[4] Chang, J., Yao, Q. and Zhou, W. (2017). Testing for high-dimensional white noise using maximum cross-correlations, Biometrika 104(1): 111–127.

[5] Chen, E. Y. and Fan, J. (2023). Statistical inference for high-dimensional matrix-variate factor model, Journal of the American Statistical Association 118: 1038–1055.

[6] Chen, E. Y., Tsay, R. S. and Chen, R. (2020). Constrained factor models for high-dimensional matrix-variate time series, Journal of the American Statistical Association 115: 775–793.

[7] Fan, J., Guo, J. and Zheng, S. (2022). Estimating number of factors by adjusted eigenvalues thresholding, Journal of the American Statistical Association 117: 852–61.

[8] Filipiak, K., Klein, D. and Vojtkova, E. (2018). The properties of partial trace and block trace operators of partitioned matrices, Electronic Journal of Linear Algebra 33: 3–15.

[9] Forni, M., Hallin, M., Lippi, M. and Reichlin, L. (2005). The generalized dynamic-factor model: one-sided estimation and forecasting, Journal of the American Statistical Association 100: 830–840.

[10] He, Y., Trapani, L., Kong, X. and Yu, L. (2023). One-way or two-way factor model for matrix sequences?, Journal of Econometrics 235: 1981–2004.

[11] Lam, C. and Yao, Q. (2012). Factor modeling for high-dimensional time series: inference for the number of factors, The Annals of Statistics 40: 694–726.

[12] Lam, C., Yao, Q. and Bathia, N. (2011). Estimation for latent factor models for highdimensional time series, Biometrika 98: 901–918.

[13] Li, Z., Lam, C., Yao, J. and Yao, Q. (2019). On testing for high-dimensional white noise, The Annals of Statistics 47(6): 3382–3412.

[14] Pan, J. and Yao, Q. (2008). Modelling multiple time series via common factors, Biometrika 95: 356–379.

[15] Wang, D., Liu, X. and Chen, R. (2019). Factor models for matrix-valued high-dimensional time series, Journal of Econometrics 208: 231–248.

[16] Yu, L., He, Y., Kong, X. and Zhang, X. (2022). Projected estimation for large-dimensional matrix factor models, Journal of Econometrics 229: 201–217.